Identification of Nonlinear Systems with Stable Limit Cycles via Convex Optimization

TL;DR

This paper introduces a convex optimization method for identifying nonlinear systems with stable limit cycles, extending existing frameworks to handle autonomous oscillations with strong stability guarantees.

Contribution

It relaxes stability constraints using transverse dynamics, enabling the identification of systems with autonomous oscillations through a convex semidefinite program.

Findings

Successfully identified a high-fidelity model of a rat hippocampal neuron

Proved simulation-error bounds without assuming true system within the model class

Established conditions for the existence of a unique limit cycle

Abstract

We propose a convex optimization procedure for black-box identification of nonlinear state-space models for systems that exhibit stable limit cycles (unforced periodic solutions). It extends the "robust identification error" framework in which a convex upper bound on simulation error is optimized to fit rational polynomial models with a strong stability guarantee. In this work, we relax the stability constraint using the concepts of transverse dynamics and orbital stability, thus allowing systems with autonomous oscillations to be identified. The resulting optimization problem is convex, and can be formulated as a semidefinite program. A simulation-error bound is proved without assuming that the true system is in the model class, or that the number of measurements goes to infinity. Conditions which guarantee existence of a unique limit cycle of the model are proved and related to the model class that we search over. The method is illustrated by identifying a high-fidelity model from experimental recordings of a live rat hippocampal neuron in culture.